RUI: Integrable Dynamics of Knotted Vortex Filaments

NSF奖摘要- DMS-0204557数学科学：RUI：可积动力学的打结涡丝摘要0204557 Calini在这个持续的项目中，我们探讨的涡丝方程，这是最简单的模型的自诱导运动的细长涡丝在理想流体中的可积性的影响，这是已知的是相当于聚焦非线性薛定谔方程（NLS）。 我们的研究的目标是相关的几何和拓扑性质-特别是结类型-大类的封闭细丝的Floquet谱相应的周期NLS解决方案，并了解如何结类型的细丝可以改变，因为它的演变。 我们采用的方法包括代数几何结构的多相解，Backlund变换，isoperiodic变形，和微扰理论。我们的工作是一个更大的趋势的一部分，使微分方程之间的联系，通常模型的物理现象，和纽结理论，这是传统上纯数学的一部分。在这里，我们研究了一个简化的模型的旋涡运动，它具有丰富的打结的循环类之间的解决方案，其数学结构是很好地理解，为了开发必要的数学工具，有效地解决问题，如结的形成，打结结构的稳定性，和结分类。

NSF Award Abstract - DMS-0204557Mathematical Sciences: RUI: Integrable Dynamics of Knotted Vortex FilamentsAbstract0204557 CaliniIn this continuing project we explore the implications of integrability for the Vortex Filament Equation, which is the simplest model of the self-induced motion of a slender vortex filament in an ideal fluid, and which is known to be equivalent to the focusing nonlinear Schrodinger equation (NLS). The goals of our research are to relate geometric and topological properties -- in particular, knot types -- of large classes of closed filaments to the Floquet spectra of the corresponding periodic NLS solutions, and to understand how the knot type of a filament can change as it evolves. The methods we employ include algebro-geometric constructions of multi-phase solutions, Backlund transformations, isoperiodic deformations, and perturbation theory. Our work is part of a larger trend of making connections between differential equations, which usually model physical phenomena, and knot theory, which is traditionally a part of pure mathematics. Here, we study a simplified model of vortex motion, which possesses a rich class of knotted loops among its solutions, and whose mathematical structure is well understood, in order to develop the mathematical tools necessary to effectively address questions such as knot formation, stability of knotted structures, and knot classification.